Question 1: Is it permissible to be a marketing expert of a company where all orders go through managers which are women with western-style clothes (open hair etc.)? (working as a marketing expert means promoting companies, shops).
Answer 1: If you are not causing, encouraging or helping them to show their hair, etc. then I can see no issue, but the question is not very clear to me.
Question 2: Is it permissible to promote selling official seals (with eagles, they are engraved as on an ordinary seal)? (I don't know exactly, but it seems to be that almost all state organisations are obliged to have such official seals)
Answer 2: According to Malik there is no issue here. If you can get a jog that is OK at least according to one of the school, then count yourself lucky and thank Allah.
Question 3: Is it permissible to promote a company as "a firm which produces package, diplomas, seals etc." (not saying exactly about official seals), if there are official seals (with an engraved eagle) in sale?
Answer 3: See above answer.
Question 4: If some percent of job means helping in haram, does it mean that the whole monthly salary is haram? Or is only the part of this salary haram?
Answer 4: If the contract has as a condition of doing haram, then it is corrupt.
... and Allah knows best.